Математика и математическое моделирование. 2019; : 1-28
Топологическая оптимизация микроструктуры адгезивов при действии тепловых и механических нагрузок
https://doi.org/10.24108/mathm.0219.0000174Аннотация
В работе излагается математическая модель и методика решения широкого класса задач топологической оптимизации адгезивного соединения для получения оптимальной микроструктуры и градиентных свойств с целью снижения уровня напряжений, возникающих за счет действия как термических, так и механических нагрузок в нем.
Адгезивные сединения имеют преимущества перед альтернативными методами соединения. В работе показано, что наиболее перспективной стратегией оптимизации адгезива является введение градуировки свойств по толщине или вдоль адгезивного слоя. Подход заключается в модификации свойств материала или геометрии адгезива, изменяющихся вдоль шва.
Во всех известных авторам работах оптимизации подлежали форма соединяемых элементов, или форма и расположение адгезивного слоя. Применение методов топологической оптимизации для определения оптимального распределения/изменения свойств градиентности самого адгезивного слоя не использовалось.
В работе проанализированы напряжения, возникающие в паяных соединениях, показано, что за счет малой толщины припоя в нем основными являются напряжения сдвига. Напряжения сдвига концентрируется вблизи концов припоя, и имеют наименьшие значения в середине. Целью задачи оптимизации является снижение пиковых значений напряжений сдвига и отслаивания в слое припоя. Топологическая оптимизация микроструктуры припоя заключается в поиске наилучшего распределения заданного количества припоя по области для достижения минимальных пиковых значений напряжений. Преимущество использования топологической оптимизации состоит в том, что микроструктура припоя не должна быть известна априори, и, таким образом, любые конструкции могут быть оптимизированы без предварительного исследования влияния исходных геометрических параметров на прочность соединения.
Алгоритм реализован на базе метода конечных элементов и метода подвижных асимптот. Рассмотрен ряд примеров с целью получения оптимальной для снижения пиковых значений напряжений сдвига и отслаивания микроструктуры припоя в трехслойном пакете.
Результаты показывают, что полученные оптимальные микроструктуры значительно снижают пиковые напряжения в слое припоя по сравнению с однородным слоем. Полученные результаты раскрывают потенциал разработанного алгоритма и показывают, что он может применяться для практических случаев.
Список литературы
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9. Hildebrand M. Non-linear analysis and optimization of adhesively bonded single lap joints between fibre-reinforced plastics and metals // Intern. J. of Adhesion and Adhesives. 1994. Vol. 14. No. 4. Pp. 261–267. DOI: 10.1016/0143-7496(94)90039-6
10. Rispler A.R., Liyong Tong, Steven G.P., Wisnom M.R. Shape optimisation of adhesive fillets // Intern. J. of Adhesion and Adhesives. 2000. Vol. 20. No. 3. Pp. 221–231. DOI: 10.1016/S0143-7496(99)00047-0
11. Taib A.A., Boukhili R., Achiou S., Gordon S., Boukehili H. Bonded joints with composite adherends. Part I. Effect of specimen configuration, adhesive thickness, spew fillet and adherend stiffness on fracture // Intern. J. of Adhesion and Adhesives. 2006. Vol. 26. No. 4. Pp. 226–236. DOI: 10.1016/j.ijadhadh.2005.03.015
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14. Haghani R., Al-Emrani M., Kliger R. Effects of geometrical modifications on behavior of adhesive joints used to bond CFRP laminates to steel members – experimental investigation // Nordic steel conf.: Nordic steel’09 (Sweden, Malmo, September 2-4, 2009): Proc. Gothenburg: Chalmers Univ. of Technology, 2009. Pp. 280-287.
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24. Ejaz H., Mubashar A., Ashcroft I.A., Uddin E., Khan M. Topology optimisation of adhesive joints using non-parametric methods // Intern. J. of Adhesion and Adhesives. 2018. Vol. 81. Pp. 1-10. DOI: 10.1016/j.ijadhadh.2017.11.003
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27. Carbas R.J.C., da Silva L.F.M., Critchlow G.W. Adhesively bonded functionally graded joints by induction heating // Intern. J. of Adhesion and Adhesives. 2014. Vol. 48. Pp. 110–118. DOI: 10.1016/j.ijadhadh.2013.09.045
28. Gojny F.H., Wichmann M.H.G., Fiedler B., Schulte K. Influence of different carbon nanotubes on the mechanical properties of epoxy matrix composites – A comparative study // Composites Science and Technology. 2005. Vol. 65. No. 15-16. Pp. 2300–2313. DOI: 10.1016/j.compscitech.2005.04.021
29. Yong Huang, Xian-Fang Li. A new approach for free vibration of axially functionally graded beams with non-uniform cross-section // J. of Sound and Vibration. 2010. Vol. 329. No. 11. Pp. 2291–2303. DOI: 10.1016/j.jsv.2009.12.029
30. Da Silva L.F.M., Adams R.D. Techniques to reduce the peel stresses in adhesive joints with composites // Intern. J. of Adhesion and Adhesives. 2007. Vol. 27. No. 3. Pp. 227-235. DOI: 10.1016/j.ijadhadh.2006.04.001
31. Hart-Smith L.J. Adhesive-bonded scarf and stepped-lap joints. Long Beach, CA: McDonnell Douglas Corp., 1973. 123 p.
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Mathematics and Mathematical Modeling. 2019; : 1-28
Topological Optimization of Microstructure of Adhesives under Thermal and Mechanical Loads
https://doi.org/10.24108/mathm.0219.0000174Abstract
The paper presents a mathematical model and method for solving a wide class of problems in topological optimization of an adhesive joint to obtain an optimal microstructure and gradient properties in order to reduce the level of stresses arising from both thermal and mechanical loads therein.
Adhesive joints have advantages over alternative bonding methods. The paper shows that the introduction of graduating properties in thickness or along the adhesive layer is the most promising strategy to optimize the adhesive. The approach is to modify the material properties or the geometry of the adhesive, varying along the joint.
In all the papers known to authors, the shape of the elements to be joined, or the shape and location of the adhesive layer, were subject to optimization. The topological optimization methods to determine the optimal distribution / change of the gradient properties of the adhesive layer itself were not used.
In the paper, the stresses arising in the solder joints are analyzed; it is shown that due to the small solder thickness, shear stresses are basic in it. The shear stresses are concentrated near the ends of the solder, and have the lowest values in the middle. The objective of the optimization problem is to reduce the peak values of the shear and peeling stresses in the solder layer. The topological optimization of the solder microstructure is to find the best distribution of a given amount of solder in the region in order to reach minimum peak values of stresses. The advantage of using topological optimization is that the microstructure of the solder should not be known a priori, and, thus, any designs can be optimized without first studying the effect of the original geometric parameters on the strength of the joint.
The algorithm is implemented using the finite element method and the method of movable asymptotes. A number of examples are considered in order to obtain the solder microstructure to be optimal for reducing the peak values of shear stresses and delamination in a three-layer package.
The results show that optimal microstructures significantly reduce peak stresses compared to a uniform layer. The obtained results reveal the potential of the developed algorithm and show that it can find practical use.
References
1. Hart-Smith L.J. Design methodology for bonded-bolted composite joints. Vol. 1: Analysis derivations and illustrative solutions. Long Beach, CA: Douglas Aircraft Co.; McDonnell Douglas Corp., 1982. 97 p.
2. Kelly G. Quasi-static strength and fatigue life of hybrid (bonded/bolted) composite single-lap joints // Composite Structures. 2006. Vol. 72. No.1. Pp. 119-129. DOI: 10.1016/j.compstruct.2004.11.002
3. Handbook of adhesion technology / Ed. by L.F.M. da Silva a.o. 2nd ed. Vol. 1-2. N.Y.: Springer, 2018. 1805 p.
4. Adams R.D., Comyn J., Wake W.C. Structural adhesive joints in engineering. 2nd ed. L.: Chapman & Hall, 1997. 359 p.
5. Dixon D.G. Adhesives used: structures bonded // Handbook of adhesion / Ed. by D.E. Packham. 2nd ed. Chichester: John Wiley & Sons, 2005.
6. Watson C. Advantages of use of adhesives: specific examples of improved design // Handbook of adhesion / Ed. by D.E. Packham. 2nd ed. Chichester: John Wiley & Sons, 2005.
7. Breto R., Chiminelli A., Duvivier E., Lizaranzu M., Jimenez M.A. Functionally graded bond-lines for metal /composite joints // 16th Eur. conf. on composite materials: ECCM’16 (Seville, Spain, June 22-26, 2014): Proc. 2014. 8 p.
8. Groth H.L., Nordlund P. Shape optimization of bonded joints // Intern. J. of Adhesion and Adhesives. 1991. Vol. 11. No. 4. Pp. 204–212. DOI: 10.1016/0143-7496(91)90002-Y
9. Hildebrand M. Non-linear analysis and optimization of adhesively bonded single lap joints between fibre-reinforced plastics and metals // Intern. J. of Adhesion and Adhesives. 1994. Vol. 14. No. 4. Pp. 261–267. DOI: 10.1016/0143-7496(94)90039-6
10. Rispler A.R., Liyong Tong, Steven G.P., Wisnom M.R. Shape optimisation of adhesive fillets // Intern. J. of Adhesion and Adhesives. 2000. Vol. 20. No. 3. Pp. 221–231. DOI: 10.1016/S0143-7496(99)00047-0
11. Taib A.A., Boukhili R., Achiou S., Gordon S., Boukehili H. Bonded joints with composite adherends. Part I. Effect of specimen configuration, adhesive thickness, spew fillet and adherend stiffness on fracture // Intern. J. of Adhesion and Adhesives. 2006. Vol. 26. No. 4. Pp. 226–236. DOI: 10.1016/j.ijadhadh.2005.03.015
12. Handbook of adhesion technology / Ed. by L.F.M. da Silva a.o. Vol. 1-2. B.; Hdbl.: Springer, 2011.
13. Da Silva L.F.M., Adams R.D. Joint strength predictions for adhesive joints to be used over a wide temperature range // Intern. J. of Adhesion and Adhesives. 2007. Vol. 27. No. 5. Pp. 362-379. DOI: 10.1016/j.ijadhadh.2006.09.007
14. Haghani R., Al-Emrani M., Kliger R. Effects of geometrical modifications on behavior of adhesive joints used to bond CFRP laminates to steel members – experimental investigation // Nordic steel conf.: Nordic steel’09 (Sweden, Malmo, September 2-4, 2009): Proc. Gothenburg: Chalmers Univ. of Technology, 2009. Pp. 280-287.
15. Lang T.P., Mallick P.K. Effect of spew geometry on stresses in single lap adhesive joints // Intern. J. of Adhesion and Adhesives. 1998. Vol. 18. No. 3. Pp.167–177. DOI: 10.1016/S0143-7496(97)00056-0
16. Belingardi G., Goglio L., Tarditi A. Investigating the effect of spew and chamfer size on the stresses in metal/plastics adhesive joints // Intern. J. of Adhesion and Adhesives. 2002. Vol. 22. No. 4. Pp. 273–282. DOI: 10.1016/S0143-7496(02)00004-0
17. Frostig Y., Thomsen O.T., Mortensen F. Analysis of adhesive-bonded joints, square-end and spew-fillet-high-order theory approach // J. of Engineering Mechanics. 1999. Vol. 125. No. 11. Pp. 1298–1307. DOI: 10.1061/(ASCE)0733-9399(1999)125:11(1298)
18. Akpinar S., Doru M.O., Ozel A., Aydin M.D., Jahanpasand H.G. The effect of the spew fillet on an adhesively bonded single-lap joint subjected to bending moment // Composites. Pt. B: Engineering. 2013. Vol. 55. Pp. 55–64. DOI: 10.1016/j.compositesb.2013.05.056
19. Zhao X., Adams R.D., da Silva L.F.M. Single lap joints with rounded adherend corners: Experimental results and strength prediction // J. of Adhesion Science and Technology. 2011. Vol. 25. No. 8. Pp. 837-856. DOI: 10.1163/016942410X520880
20. Zhao X., Adams R.D., da Silva L.F.M. Single lap joints with rounded adherend corners: Stress and strain analysis // J. of Adhesion Science and Technology. 2011. Vol. 25. No. 8. Pp. 819-836. DOI: 10.1163/016942410X520871
21. Matveenko V.P., Sevodina N.V., Fedorov A.Yu. Optimizatsiya geometrii uprugikh tel v okrestnostyakh osobykh tochek na primere kleevogo soedineniya vnakhlestku // Prikladnaya mekhanika i tekhnicheskaya fizika. 2013. T. 54. № 5. S. 180-186.
22. Sancaktar E., Simmons S.R. Optimization of adhesively-bonded single lap joints by adherend notching // J. of Adhesion Science and Technology. 2000. Vol. 14. No. 11. Pp.1363–1404. DOI: 10.1163/156856100742258
23. Kaye R.H., Heller M. Through-thickness shape optimisation of bonded repairs and lap-joints // Intern. J. of Adhesion and Adhesives. 2002. Vol. 22. No. 1. Pp. 7-21. DOI: 10.1016/S0143-7496(01)00029-X
24. Ejaz H., Mubashar A., Ashcroft I.A., Uddin E., Khan M. Topology optimisation of adhesive joints using non-parametric methods // Intern. J. of Adhesion and Adhesives. 2018. Vol. 81. Pp. 1-10. DOI: 10.1016/j.ijadhadh.2017.11.003
25. Stapleton S.E., Waas A.M.,Bednarcyk B.A. Modeling progressive failure of bonded joints using a single joint finite element // AIAA J. 2011. Vol. 49. No. 8. Pp. 1740-1749. DOI: 10.2514/1.J050889
26. Hsieh T.H., Kinloch A.J., Masania K., Taylor A.C., Sprenger S. The mechanisms and mechanics of the toughening of epoxy polymers modified with silica nanoparticles // Polymer. 2010. Vol. 51. No. 26. Pp. 6284-6294. DOI: 10.1016/j.polymer.2010.10.048
27. Carbas R.J.C., da Silva L.F.M., Critchlow G.W. Adhesively bonded functionally graded joints by induction heating // Intern. J. of Adhesion and Adhesives. 2014. Vol. 48. Pp. 110–118. DOI: 10.1016/j.ijadhadh.2013.09.045
28. Gojny F.H., Wichmann M.H.G., Fiedler B., Schulte K. Influence of different carbon nanotubes on the mechanical properties of epoxy matrix composites – A comparative study // Composites Science and Technology. 2005. Vol. 65. No. 15-16. Pp. 2300–2313. DOI: 10.1016/j.compscitech.2005.04.021
29. Yong Huang, Xian-Fang Li. A new approach for free vibration of axially functionally graded beams with non-uniform cross-section // J. of Sound and Vibration. 2010. Vol. 329. No. 11. Pp. 2291–2303. DOI: 10.1016/j.jsv.2009.12.029
30. Da Silva L.F.M., Adams R.D. Techniques to reduce the peel stresses in adhesive joints with composites // Intern. J. of Adhesion and Adhesives. 2007. Vol. 27. No. 3. Pp. 227-235. DOI: 10.1016/j.ijadhadh.2006.04.001
31. Hart-Smith L.J. Adhesive-bonded scarf and stepped-lap joints. Long Beach, CA: McDonnell Douglas Corp., 1973. 123 p.
32. Oterkus E., Barut A., Madenci E., Smeltzer S.S., Ambur D.R. Nonlinear analysis
33. of bonded composite single-lap joints // 45th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics & materials conf. (Palm Springs, CA, USA, April 19-22, 2004): Proc. Wash.: AIAA, 2004. 18 p. DOI: 10.2514/6.2004-1560
34. Boss J.N., Ganesh V.K., Lim C.T. Modulus grading versus geometrical grading of composite adherends in single-lap bonded joints // Composite Structures. 2003. Vol. 62. No. 1. Pp. 113–121. DOI: 10.1016/S0263-8223(03)00097-7
35. Kumar S. Analysis of tubular adhesive joints with a functionally modulus graded bondline subjected to axial loads // Intern. J. of Adhesion and Adhesives. 2009. Vol. 29. No. 8. Pp. 785-795. DOI: 10.1016/j.ijadhadh.2009.06.006
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